One of the instruments to improve the competitiveness of higher education establishments and to enhance the quality of faculty staff is a competitive selection of candidates to fill teaching positions. The purpose of the study is to analyze the effectiveness of the competitive procedure carried out in a large Russian university and to find ways to improve it as a tool for increasing the staff potential of higher schools. A new system of evaluation of educational and research activities of applicants for teaching positions has been tested for a year. Then the authors conducted a survey among the institutes’ competitive commissions members who take part in one of the competition stages. The survey revealed the advantages and disadvantages of the university’s competitive procedure. Among the main problems are the insufficient number of alternative competitions, the difficulty of attracting new candidates to the competition, the desire of department staff to choose the candidates they have been working with even if they show low-productivity, and the lack of a number of important evaluation indicators. On the basis of the study of theoretical materials and practical experience at the university, as well as the analysis of the practice of other higher educational establishments, the authors propose ways to improve the competition procedure for faculty staff. The study contributes to the search for solutions of one of the most discussed problems of attracting new teachers to the university and selecting qualified personnel with deep professional knowledge, significant scientific achievements in the relevant branches of science, scientific potential and the intellectual and moral qualities.